Creating an eye/eyelid rig for dragon without shape keys
Concerning the eye movement, you can use a UVWarp modifier to your eye.



The principle is to set two objects (empties) to shift the position of the UV map of the eye.
Moving (or inserting key frames) one empty will shift the UV map to obtain the result.

Here is a closer view of the modifier :


Edit :
Concerning the eyelid animation, if you set the bone origine at the same position your 3D cursor is (as you said you can rotate it manually), that may work.



Another edit : a rigged version of the eye animation.

The principle is simply to add a "Bone" and define a group named "Bone" for the Iris circle. That allow to play on the position and X-scale of the iris.



The Bone group is the iris / center UVs in the image below :